Unconstrained receding-horizon control of nonlinear systems

It is well known that unconstrained infinite-horizon optimal control may be used to construct a stabilizing controller for a nonlinear system. In this note, we show that similar stabilization results may be achieved using unc onstrained finite horizon optimal control, The key idea is to approximate t he tail of the infinite horizon cost-to-go using, as terminal cost, an appr opriate control Lyapunov function. Roughly speaking, the terminal control L yapunov function (CLF) should provide an (incremental) upper bound on the c ost. In this fashion, important stability characteristics may be retained w ithout the use of terminal constraints such as those employed by a number o f other researchers, The absence of constraints allows a significant speedu p in computation, Furthermore, it is shown that in order to guarantee stabi lity, it suffices to satisfy an improvement property, thereby relaxing the requirement that truly optimal trajectories be found, We provide a complete analysis of the stability and region of attraction/operation properties of receding horizon control strategies that utilize finite horizon approximat ions in the proposed class. It is shown that the guaranteed region of opera tion contains that of the CLF controller and may be made as large as desire d by increasing the optimization horizon (restricted, of course, to the inf inite horizon domain). Moreover, it is easily seen that both CLF and infini te-horizon optimal control approaches are limiting cases of our receding ho rizon strategy. The key results are illustrated using a familiar example, the inverted pend ulum, where significant improvements in guaranteed region of operation and cost are noted.